Joy of Mathematics

Course No. 1411
Professor Arthur T. Benjamin, Ph.D.
Harvey Mudd College
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What Will You Learn?

  • Reinvigorate your love of mathematics, and discover how the field can be enjoyed for its beauty and certainty.
  • Learn the beautiful and unexpected properties of Fibonacci numbers, which show up in art, nature, and even poetry.
  • Explore the bizarre concept of infinity, and deliberate over whether there are an infinite number of infinities.
  • Examine the concepts of imaginary and complex numbers, which play a surprisingly useful role in physics.
  • Study how mathematical proofs work, including proofs by deduction, proofs by induction, and proofs by existence.

Course Overview

Ready to exercise those brain cells? Humans have been having fun with mathematics for thousands of years. Along the way, they've discovered the amazing utility of this field—in science, engineering, finance, games of chance, and many other aspects of life. This course of 24 half-hour lectures celebrates the sheer joy of mathematics, taught by a mathematician who is literally a magician with numbers. Professor Arthur T. Benjamin of Harvey Mudd College is renowned for his feats of mental calculation performed before audiences at schools, theaters, museums, conferences, and other venues.

Although racing a calculator to solve a difficult problem may seem like a superhuman achievement, Professor Benjamin shows that there are simple tricks that allow anyone to look like a math magician. Professor Benjamin has another goal in this course: throughout these lectures, he shows how everything in mathematics is connected—how the beautiful and often imposing edifice that has given us algebra, geometry, trigonometry, calculus, probability, and so much else is based on nothing more than fooling around with numbers.

Fun with Numbers

Here is an example:

Think of a number between 1 and 10. Triple it. Add 6. Then triple again. Now take your answer, probably a two-digit number, and add the digits of your answer. If you still have a two-digit number, add those digits again. You should now be thinking of the magical number 9. The reason this works is based on algebra and the fact that the digits of any multiple of 9 must sum to a multiple of 9.

This is one of the many wonders of modular arithmetic, sometimes called clock arithmetic, where numbers wrap around in a circle. A useful application of this field is casting out nines, a simple and ancient technique for checking the answers to arithmetical problems.

Modular arithmetic also provides a very handy method for mentally computing the day of the week for any date in history.

This connection between entertaining number tricks and the deeper properties of mathematics reflects Dr. Benjamin's specialty, which is combinatorics, the branch of mathematics that deals with the subtleties of counting. Some examples: How many different six-symbol license plates are possible? And for the book collector, how many ways are there of arranging 10 books on a shelf? (Would you believe more than 3 million?) These simple questions introduce concepts such as the factorial function.

Drawing on his dual fascination with combinatorics and games, Dr. Benjamin used his analytical skill to win first place in the American Backgammon Tour in 1997.

Have You Forgotten Math? Worry Not!

Professor Benjamin gives his presentation on the number 9 in—where else?—Lecture 9. Other lectures are devoted to pi, the imaginary number i, the transcendental number e, and infinity. These numbers are gateways to intriguing realms of mathematics, which you explore under Dr. Benjamin's enthusiastic guidance.

He also introduces you to prime numbers, Fibonacci numbers, and infinite series. And you investigate the powerful techniques for manipulating numbers using algebra, geometry, trigonometry, calculus, and probability in lectures that may hark back to subjects you studied in high school and college. You will find Dr. Benjamin's introduction to these fields both a useful refresher and a bird's-eye view of the most important areas of mathematics. Intriguingly, he approaches these topics from the novel perspective of combinatorics and mathematical games, providing a fun entry into subjects that are often taught in a lackluster way.

In the next-to-last lecture, you look at the application of probability to games. And finally, you splurge on feats of mathematical magic: For instance, did you know that you don't have to be a genius to calculate cube roots in your head?

Throughout the course, Dr. Benjamin assumes that you may have no more than a distant memory of high school math. He believes that it is his job to fan those embers into a burning interest in the subject he loves so much—and in which he takes such exquisite joy.

A Math Course Designed for You

This course is especially well suited for:

  • Anyone attracted by Dr. Benjamin's promise of a joyful attitude to an often-imposing subject
  • Anyone for whom high school and college math courses are a distant memory who would like to revisit these subjects to explore topics they skipped the first time
  • Anyone now taking math who would like a big-picture perspective on the major areas of the field from a playful, joyous point of view
  • Budding math mavens who love numbers and the magic that can be done with them.

Be prepared to encounter strange equations, novel ways of thinking, and symbols and computational methods that may be new to you. But also prepare to sharpen your wits in ways you never thought possible. Math is a challenging subject, but it pays immense rewards. Few people understand everything the first time through an unfamiliar domain of math. "But that's OK," says Dr. Benjamin. "You can rewind me and have me explain it all over again! All of this material bears repeating, and I hope you get to enjoy it many times over."

Patterns, Patterns Everywhere

One of Dr. Benjamin's greatest loves is the Fibonacci sequence, which shows up in many spheres of mathematics, as well as in nature, art, computer science, and poetry. The distinctive meter of a limerick encodes Fibonacci numbers, and Dr. Benjamin has even composed his own limerick to show how the sequence begins:

I think Fibonacci is fun;
We start with a 1 and a 1.
Then 2, 3, 5, 8,
But don't stop there, mate!
The fun has just barely begun.

The series continues on: 13, 21, 34, 55, 89, … with each successive Fibonacci number being the sum of the previous two. This simple pattern is named for a 12th-century mathematician who described a problem involving imaginary rabbits that never die. Starting with a pair of baby rabbits, the animals take a month to mature, then mate and produce a male and a female; these mature after a month and mate, along with their parents. The total number of pairs after each month follows the Fibonacci sequence.

In our own day, Fibonacci numbers appear as a critical plot element in The Da Vinci Code, notably under the guise of the golden ratio, an ideal proportion favored by artists and architects that is intimately connected to the Fibonacci sequence. However, Dr. Benjamin cautions that the quest for instances of the golden ratio in nature can get out of hand.

A Feast for the Brain

Anyone who has ever witnessed a feat of mathematical prowess and chalked it up to unfathomable intellect will be interested to learn that there is often a simple shortcut at work. For example, the first 24 digits of pi, the famous ratio of the circumference of a circle to its diameter, can be memorized with the help of a silly sentence starting "My turtle Pancho …" Four more sentences take you to 100 digits, making you look like a prodigy indeed!

Similarly, if someone asked you to add up all the numbers from 1 to 100, you might take out a sheet of paper and start to work, little realizing that this difficult-looking problem can be done in your head in seconds by a method devised by the famous mathematician Carl Friedrich Gauss when he was a boy. The same goes for squaring and multiplying multidigit numbers in your head.

"Mathematics is food for the brain," says Dr. Benjamin. "It helps you think precisely, decisively, and creatively and helps you look at the world from multiple perspectives. Naturally, it comes in handy when you're shopping around for the best bargain or trying to understand the statistics you read in the newspaper.

"But I hope that you come away from this course with a new way to experience beauty—in the form of a surprising pattern or an elegant logical argument. Many people find joy in fine music, poetry, and other works of art—and mathematics offers joys that I hope you, too, will learn to experience. If Elizabeth Barrett Browning had been a mathematician, she might have said, ‘How do I count thee? Let me love the ways!'"

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24 lectures
 |  Average 30 minutes each
  • 1
    The Joy of Math—The Big Picture
    Professor Benjamin introduces the ABCs of math appreciation: The field can be loved for its applications, its beauty and structure, and its certainty. Most of all, mathematics is a source of endless delight through creative play with numbers. x
  • 2
    The Joy of Numbers
    How do you add all the numbers from 1 to 100—instantly? What makes a square number square and a triangular number triangular? Why do the rules of arithmetic really work, and how do you calculate in bases other than 10? x
  • 3
    The Joy of Primes
    A number is prime if it is evenly divisible by only itself and one: for example, 2, 3, 5, 7, 11. Professor Benjamin proves that there are an infinite number of primes and shows how they are the building blocks of our number system. x
  • 4
    The Joy of Counting
    Combinatorics is the study of counting questions such as: How many outfits are possible if you own 8 shirts, 5 pairs of pants, and 10 ties? A trickier question: How many ways are there to arrange 10 books on a shelf? Combinatorics can also be used to analyze numbering systems, such as ZIP Codes or license plates, as well as games of chance. x
  • 5
    The Joy of Fibonacci Numbers
    The Fibonacci numbers follow the simple pattern 1, 1, 2, 3, 5, 8, etc., in which each number is the sum of the two preceding numbers. Fibonacci numbers have many beautiful and unexpected properties, and show up in nature, art, and poetry. x
  • 6
    The Joy of Algebra
    Arguably the most important area of mathematics, algebra introduces the powerful idea of using an abstract variable to represent an unknown quantity. This lecture demonstrates algebra's golden rule: Do unto one side of an equation as you do unto the other. x
  • 7
    The Joy of Higher Algebra
    This lecture shows how to solve quadratic (second-degree) equations from the technique of completing the square and the quadratic formula. The quadratic formula reveals the connection between Fibonacci numbers and the golden ratio. x
  • 8
    The Joy of Algebra Made Visual
    Algebra can be used to solve geometrical problems, such as finding where two lines cross. The technique is useful in real-life problems, for example, in choosing a telephone plan. Graphs help us better understand everything from lines to equations with negative or fractional exponents. x
  • 9
    The Joy of 9
    Adding the digits of a multiple of 9 always gives a multiple of 9. For example: 9 x 4 = 36, and 3 + 6 = 9. In modular arithmetic, this property allows checking answers by "casting out nines." A related trick: mentally computing the day of the week for any date in history. x
  • 10
    The Joy of Proofs
    Professor Benjamin begins his discussion of mathematical proofs with intuitive cases like "even plus even is even" and "odd times odd is odd." He builds to more complex proofs by existence and induction, and ends with a checkerboard challenge. x
  • 11
    The Joy of Geometry
    Geometry is based on a handful of definitions and axioms involving points, lines, and angles. These lead to important conclusions about the properties of polygons. This lecture uses geometric reasoning to derive the Pythagorean theorem and other interesting results. x
  • 12
    The Joy of Pi
    Pi is the ratio of the circumference of a circle to its diameter. It starts 3.14 and continues in an infinite nonrepeating sequence. Professor Benjamin shows how to learn the first hundred digits of this celebrated number, making it look as easy as pie. x
  • 13
    The Joy of Trigonometry
    Trigonometry deals with the sides and angles of triangles. This lecture defines sine, cosine, and tangent, along with their reciprocals, the cosecant, secant, and cotangent. Extending these definitions to the unit circle allows a handy measure of angle: the radian. x
  • 14
    The Joy of the Imaginary Number i
    Could the apparently nonsensical number the square root of –1 be of any use? Very much so, as this lecture shows. Such imaginary and complex numbers play an indispensable role in physics and other fields, and are easier to understand than they appear. x
  • 15
    The Joy of the Number e
    Another indispensable number to learn is e = 2.71828 ... Defined as the base of the natural logarithm, e plays a central role in calculus, and it arises naturally in many spheres of mathematics, including calculations of compound interest. x
  • 16
    The Joy of Infinity
    What is the meaning of infinity? Are some infinite sets "more" infinite than others? Could there possibly be an infinite number of levels of infinity? This lecture explores some of the strange ideas associated with mathematical infinity. x
  • 17
    The Joy of Infinite Series
    Starting with the analysis of the proposition 0.999999999 ... = 1, this lecture ex­plores what it means to add up an infinite series of numbers. Some infinite series con­verge on a definite value, while others grow arbitrarily large. x
  • 18
    The Joy of Differential Calculus
    Calculus is the mathematics of change, and answers questions such as: How fast is a function growing? This lecture introduces the concepts of limits and derivatives, which allow the slope of a curve to be measured at any point. x
  • 19
    The Joy of Approximating with Calculus
    Exploiting the idea of the derivative, we can approximate just about any function using simple polynomials. This lecture also shows why a formula sometimes known as "God's equation" (involving e, i, p, 1, and 0) is true, and how to calculate square roots in your head. x
  • 20
    The Joy of Integral Calculus
    Geometry and trigonometry are used to determine the areas of simple figures such as triangles and circles. But how are more complex shapes measured? Calculus comes to the rescue with a technique called integration, which adds the simple areas of many tiny quantities. x
  • 21
    The Joy of Pascal's Triangle
    A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of beautiful number patterns. It even provides an answer to the song "The Twelve Days of Christmas": Exactly how many gifts did my true love give to me? x
  • 22
    The Joy of Probability
    Mathematics can draw detailed inferences about random events. This lecture covers major concepts in probability, such as the law of large numbers, the central limit theorem, and how to measure variance. x
  • 23
    The Joy of Mathematical Games
    This lecture applies the law of total probability and other concepts from the course to predict the long-term losses to be expected from playing games such as roulette and craps and understand what is known as the "Gambler's Ruin Problem." x
  • 24
    The Joy of Mathematical Magic
    Closing the course with a magician's flair, Professor Benjamin shows a trick for producing anyone's phone number, how to create a magic square based on your birthday, how to play "mathematical survivor," a technique for computing cube roots in your head, and a card trick to ponder. x

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Your professor

Arthur T. Benjamin

About Your Professor

Arthur T. Benjamin, Ph.D.
Harvey Mudd College
Dr. Arthur T. Benjamin is Professor of Mathematics at Harvey Mudd College. He earned a Ph.D. in Mathematical Sciences from Johns Hopkins University in 1989. Professor Benjamin's teaching has been honored repeatedly by the Mathematical Association of America (MAA). In 2000, he received the MAA Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics. The MAA also named...
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Reviews

Joy of Mathematics is rated 4.4 out of 5 by 168.
Rated 2 out of 5 by from disappointing so far Mathematics is the language of science. I had expected to learn about how math is used to solve interesting problems in science and technology. So far, lecture 7, this is completely lacking. Benjamin uses algebra to play with numbers. Big whoop! I don't care. I can multiply large numbers longhand. He plays with fibonacci numbers but never mentions their importance in nature. Benjamin plays with math like a toy. I use math to solve real problems. I did better in high school. So far I'm very disappointed. Trevor Smith
Date published: 2018-10-20
Rated 4 out of 5 by from A few complaints, but worth it for some I am a retired engineer and still tinker with Excel, personal finance and regularly purchase math and logic problem puzzle magazines. So, I am not afraid of numbers. I think I was drawn into the course by the title, thinking that there would be something new or different included. I think the course is more or less a survey of high school and first year college math classes. In the end, this course was a refresher for me, particularly in the areas that turned out to be useless for me both at work and everyday living. As I watched this course, I just kept on thinking to myself, “no wonder so many people either hate math or just don’t ever get it.! There were so many equations, theorems, proofs, and mathematical jargon thrown out, with just a little bit of reference to real life applications. Sure, now and again he talks about applications (calculating odds in poker and horse racing, calculating interest rates,, etc.) He also mixes in some tricks for mental math or entertaining audiences, but I don’t think they will ever be that useful for me. Some of the tricks were very easy (such as multiplying any number by 11 or squaring any number ending in a 5), but I have a hard time figuring out when I will ever find the need to use them, maybe in some of the puzzles I do? In other cases, these “tricks” were quite convoluted and it just seems easier to stick with the traditional way you learned how to do things in school. This professor is clearly enthusiastic about this topic and knows his numbers. He is pretty animated and talks with his hands (and arms) quite a bit. There are times where he just zips through a topic, while other times it was just painfully slow. He spends a lot of time proclaiming that numbers and their patterns are beautiful, great, awesome, etc. I wish he would spend that time talking about real life applications of the math concepts, the importance or usefulness (rather than the beauty) of number patterns, and why it is important to go through the process of proving theorems (i.e. is it more than something you need to know to pass a test?) In other words, take it out of an academic mindset! Otherwise, it kind of leaves you thinking that math is mostly about professors and grad students spending every minute of the day looking for numerical patterns or the next largest prime number (which somehow will make you very famous). These real world examples don’t have to be shown through detailed mathematical calculations, but with quite a bit more discussion than briefly mentioning that Fibonacci numbers are used by biologists or that you find the imaginary number in engineering. I think this would help draw more people into the concept of “joy of math”. For me, the only ‘joy” in this course is the magic tricks that may be interesting to some, particularly children and teenagers. Even with the shortcomings, this course is useful. The professor does get the general points across in a variety of areas of math, and if enthusiasm helps you learn, this guy will give you a good dose of that! Generally, this course gives you kind of a primer if your are taking more challenging high school or college level math classes. If you have been out of school for awhile and are familiar with math, this will be a good refresher for you. Just be ready to pause and rewind from time to time. However,It is not a deep dive in any particular area. Also, if you are intimidated by math and do not need the higher level of math discussed (see the list of lectures to see what is covered), I doubt you will find this course useful.
Date published: 2018-10-04
Rated 5 out of 5 by from Good way to get excited about learning math I bought this course a few years ago and set it aside until I finally got around to my resolution about actually LEARNING mathematics. I've finished Algebra I and II (Great Courses, Prof. Sellers) and Geometry (Great Course, Prof. Tanton) and decided to listen to these lectures to fire myself up (further) for Precalculus. A couple points: 1) You do need to be familiar with some mathematics to be able to appreciate what Professor Benjamin is teaching. You don't need to KNOW it - just have some familiarity. He gets into a surprising amount of depth and it simply will not make any sense at all unless you have some algebra and geometry. 2) Don't buy this course if you think it will teach you math. It does not - nor was it intended to. Its function is to introduce you to what math is, how much fun and interesting it can be and what you can learn if you are willing to wade in deeper and actually do the work. If you keep that in mind, you'll have a really good time listening to Professor Benjamin. He has the characteristic that all the best professors have - the reason why the Great Courses professors are so good: he is absolutely delighted by his subject and he communicates that delight and passion to you in a way that not only makes you want to join that "club" but makes you believe you can. He tells you about each of the subjects he has chosen and includes some fun math "tricks" in each lecture. For once, the lecture notes that come in the accompanying booklet are thorough enough so that you can actually go back to the "tricks" with enough information to practice them. If I were to change anything about the lectures, it would simply be to add a few examples of real life professions made possible by the work on the subject of the particular lecture. For example, the invention of imaginary numbers made it possible for . . . . But that's not a criticism really. Figuring it out is part of the fun of the subject.
Date published: 2018-09-13
Rated 5 out of 5 by from Joy to Mathematicians and Magicians I’m both, so was interested in what this was about. I enjoyed it immensely, especially the magic. I think I’ll have my granddaughter view some of it while she is on vacation with me in a few weeks. She just started in magic so this should peak her interest in math as well, even at age 12 she’s had Algebra, Trig, and Geometry. I found only one place where instructor assumed more than a student might know.
Date published: 2018-07-20
Rated 5 out of 5 by from Great! I bought this for my daughter who is going into high school. She LOVES math and is very exited to have this corse.
Date published: 2018-07-16
Rated 5 out of 5 by from Joy of Mathematics These are topics I explore with my Middle School students. Since Professor Benjamin is a much better expert than I am, I will now be able to add to my lessons.
Date published: 2018-07-13
Rated 5 out of 5 by from Really fun math I am writing a review & haven't even finished the course yet. So far, thoroughly enjoying this class. The professor explains his concepts and is really fun to watch on video. It is a shame that math isn't taught this way in schools because I believe more children would be into this. Super job Professor! Cannot wait to finish & look forward to my daily lessons!
Date published: 2018-05-23
Rated 5 out of 5 by from Well named course! I haven’t taken a math class since high school, so I decided to take this course to see if I would still "hit the wall" after algebra. I've always felt that I didn't understand because I wasn't taught the practical applications of the math, so I went through the early lectures still wondering in what situations one would use the things he was talking about. When I got into the second half of the course, I realized the early lectures were intended to cover some amazing properties of numbers - primes, Fibonacci numbers, pi, etc., have a way of just showing up in the results of various formulas in more advanced math. Professor Benjamin did a great job of explaining the practical applications, so to my great joy, I actually understood not only how to do the math, but where and why I might need it. As a retiree, my need for the math may be somewhat limited, but it does exist and I will likely go back through the lectures again (and maybe even more than once). I would imagine this course would be of even greater use to a young student, to supplement junior and senior high school math courses and to encourage further study. I most definitely recommend this course. It was most appropriately named! I thought Professor Benjamin was an interesting lecturer - he kind of fits the stereotype of a mathematician, very bright and just a little goofy. He covered a lot in comparatively few lectures and was very clear and concise. I would definitely take another course from him.
Date published: 2018-04-28
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